1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 22 /* 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 24 */ 25 /* 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 /* 31 * floating point Bessel's function of the first and second kinds 32 * of order zero: j1(x),y1(x); 33 * 34 * Special cases: 35 * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; 36 * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. 37 */ 38 39 #pragma weak __j1 = j1 40 #pragma weak __y1 = y1 41 42 #include "libm.h" 43 #include "libm_protos.h" 44 #include <math.h> 45 #include <values.h> 46 47 #define GENERIC double 48 static const GENERIC 49 zero = 0.0, 50 small = 1.0e-5, 51 tiny = 1.0e-20, 52 one = 1.0, 53 invsqrtpi = 5.641895835477562869480794515607725858441e-0001, 54 tpi = 0.636619772367581343075535053490057448; 55 56 static GENERIC pone(GENERIC), qone(GENERIC); 57 static const GENERIC r0[4] = { 58 -6.250000000000002203053200981413218949548e-0002, 59 1.600998455640072901321605101981501263762e-0003, 60 -1.963888815948313758552511884390162864930e-0005, 61 8.263917341093549759781339713418201620998e-0008, 62 }; 63 static const GENERIC s0[7] = { 64 1.0e0, 65 1.605069137643004242395356851797873766927e-0002, 66 1.149454623251299996428500249509098499383e-0004, 67 3.849701673735260970379681807910852327825e-0007, 68 }; 69 static const GENERIC r1[12] = { 70 4.999999999999999995517408894340485471724e-0001, 71 -6.003825028120475684835384519945468075423e-0002, 72 2.301719899263321828388344461995355419832e-0003, 73 -4.208494869238892934859525221654040304068e-0005, 74 4.377745135188837783031540029700282443388e-0007, 75 -2.854106755678624335145364226735677754179e-0009, 76 1.234002865443952024332943901323798413689e-0011, 77 -3.645498437039791058951273508838177134310e-0014, 78 7.404320596071797459925377103787837414422e-0017, 79 -1.009457448277522275262808398517024439084e-0019, 80 8.520158355824819796968771418801019930585e-0023, 81 -3.458159926081163274483854614601091361424e-0026, 82 }; 83 static const GENERIC s1[5] = { 84 1.0e0, 85 4.923499437590484879081138588998986303306e-0003, 86 1.054389489212184156499666953501976688452e-0005, 87 1.180768373106166527048240364872043816050e-0008, 88 5.942665743476099355323245707680648588540e-0012, 89 }; 90 91 GENERIC 92 j1(GENERIC x) { 93 GENERIC z, d, s, c, ss, cc, r; 94 int i, sgn; 95 96 if (!finite(x)) 97 return (one/x); 98 sgn = signbit(x); 99 x = fabs(x); 100 if (x > 8.00) { 101 s = sin(x); 102 c = cos(x); 103 /* 104 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0)) 105 * where x0 = x-3pi/4 106 * Better formula: 107 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 108 * = 1/sqrt(2) * (sin(x) - cos(x)) 109 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 110 * = -1/sqrt(2) * (cos(x) + sin(x)) 111 * To avoid cancellation, use 112 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 113 * to compute the worse one. 114 */ 115 if (x > 8.9e307) { /* x+x may overflow */ 116 ss = -s-c; 117 cc = s-c; 118 } else if (signbit(s) != signbit(c)) { 119 cc = s - c; 120 ss = cos(x+x)/cc; 121 } else { 122 ss = -s-c; 123 cc = cos(x+x)/ss; 124 } 125 /* 126 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss) 127 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc) 128 */ 129 if (x > 1.0e40) 130 d = (invsqrtpi*cc)/sqrt(x); 131 else 132 d = invsqrtpi*(pone(x)*cc-qone(x)*ss)/sqrt(x); 133 134 if (x > X_TLOSS) { 135 if (sgn != 0) { d = -d; x = -x; } 136 return (_SVID_libm_err(x, d, 36)); 137 } else 138 if (sgn == 0) 139 return (d); 140 else 141 return (-d); 142 } 143 if (x <= small) { 144 if (x <= tiny) 145 d = 0.5*x; 146 else 147 d = x*(0.5-x*x*0.125); 148 if (sgn == 0) 149 return (d); 150 else 151 return (-d); 152 } 153 z = x*x; 154 if (x < 1.28) { 155 r = r0[3]; 156 s = s0[3]; 157 for (i = 2; i >= 0; i--) { 158 r = r*z + r0[i]; 159 s = s*z + s0[i]; 160 } 161 d = x*0.5+x*(z*(r/s)); 162 } else { 163 r = r1[11]; 164 for (i = 10; i >= 0; i--) r = r*z + r1[i]; 165 s = s1[0]+z*(s1[1]+z*(s1[2]+z*(s1[3]+z*s1[4]))); 166 d = x*(r/s); 167 } 168 if (sgn == 0) 169 return (d); 170 else 171 return (-d); 172 } 173 174 static const GENERIC u0[4] = { 175 -1.960570906462389461018983259589655961560e-0001, 176 4.931824118350661953459180060007970291139e-0002, 177 -1.626975871565393656845930125424683008677e-0003, 178 1.359657517926394132692884168082224258360e-0005, 179 }; 180 static const GENERIC v0[5] = { 181 1.0e0, 182 2.565807214838390835108224713630901653793e-0002, 183 3.374175208978404268650522752520906231508e-0004, 184 2.840368571306070719539936935220728843177e-0006, 185 1.396387402048998277638900944415752207592e-0008, 186 }; 187 static const GENERIC u1[12] = { 188 -1.960570906462389473336339614647555351626e-0001, 189 5.336268030335074494231369159933012844735e-0002, 190 -2.684137504382748094149184541866332033280e-0003, 191 5.737671618979185736981543498580051903060e-0005, 192 -6.642696350686335339171171785557663224892e-0007, 193 4.692417922568160354012347591960362101664e-0009, 194 -2.161728635907789319335231338621412258355e-0011, 195 6.727353419738316107197644431844194668702e-0014, 196 -1.427502986803861372125234355906790573422e-0016, 197 2.020392498726806769468143219616642940371e-0019, 198 -1.761371948595104156753045457888272716340e-0022, 199 7.352828391941157905175042420249225115816e-0026, 200 }; 201 static const GENERIC v1[5] = { 202 1.0e0, 203 5.029187436727947764916247076102283399442e-0003, 204 1.102693095808242775074856548927801750627e-0005, 205 1.268035774543174837829534603830227216291e-0008, 206 6.579416271766610825192542295821308730206e-0012, 207 }; 208 209 210 GENERIC 211 y1(GENERIC x) { 212 GENERIC z, d, s, c, ss, cc, u, v; 213 int i; 214 215 if (isnan(x)) 216 return (x*x); /* + -> * for Cheetah */ 217 if (x <= zero) { 218 if (x == zero) 219 /* return -one/zero; */ 220 return (_SVID_libm_err(x, x, 10)); 221 else 222 /* return zero/zero; */ 223 return (_SVID_libm_err(x, x, 11)); 224 } 225 if (x > 8.0) { 226 if (!finite(x)) 227 return (zero); 228 s = sin(x); 229 c = cos(x); 230 /* 231 * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x0)-q1(x)*sin(x0)) 232 * where x0 = x-3pi/4 233 * Better formula: 234 * cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4) 235 * = 1/sqrt(2) * (sin(x) - cos(x)) 236 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) 237 * = -1/sqrt(2) * (cos(x) + sin(x)) 238 * To avoid cancellation, use 239 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) 240 * to compute the worse one. 241 */ 242 if (x > 8.9e307) { /* x+x may overflow */ 243 ss = -s-c; 244 cc = s-c; 245 } else if (signbit(s) != signbit(c)) { 246 cc = s - c; 247 ss = cos(x+x)/cc; 248 } else { 249 ss = -s-c; 250 cc = cos(x+x)/ss; 251 } 252 /* 253 * j1(x) = 1/sqrt(pi*x) * (P(1,x)*cc - Q(1,x)*ss) 254 * y1(x) = 1/sqrt(pi*x) * (P(1,x)*ss + Q(1,x)*cc) 255 */ 256 if (x > 1.0e91) 257 d = (invsqrtpi*ss)/sqrt(x); 258 else 259 d = invsqrtpi*(pone(x)*ss+qone(x)*cc)/sqrt(x); 260 261 if (x > X_TLOSS) 262 return (_SVID_libm_err(x, d, 37)); 263 else 264 return (d); 265 } 266 if (x <= tiny) { 267 return (-tpi/x); 268 } 269 z = x*x; 270 if (x < 1.28) { 271 u = u0[3]; v = v0[3]+z*v0[4]; 272 for (i = 2; i >= 0; i--) { 273 u = u*z + u0[i]; 274 v = v*z + v0[i]; 275 } 276 } else { 277 for (u = u1[11], i = 10; i >= 0; i--) u = u*z+u1[i]; 278 v = v1[0]+z*(v1[1]+z*(v1[2]+z*(v1[3]+z*v1[4]))); 279 } 280 return (x*(u/v) + tpi*(j1(x)*log(x)-one/x)); 281 } 282 283 static const GENERIC pr0[6] = { 284 -.4435757816794127857114720794e7, 285 -.9942246505077641195658377899e7, 286 -.6603373248364939109255245434e7, 287 -.1523529351181137383255105722e7, 288 -.1098240554345934672737413139e6, 289 -.1611616644324610116477412898e4, 290 }; 291 static const GENERIC ps0[6] = { 292 -.4435757816794127856828016962e7, 293 -.9934124389934585658967556309e7, 294 -.6585339479723087072826915069e7, 295 -.1511809506634160881644546358e7, 296 -.1072638599110382011903063867e6, 297 -.1455009440190496182453565068e4, 298 }; 299 static const GENERIC huge = 1.0e10; 300 301 static GENERIC 302 pone(GENERIC x) { 303 GENERIC s, r, t, z; 304 int i; 305 /* assume x > 8 */ 306 if (x > huge) 307 return (one); 308 309 t = 8.0/x; z = t*t; 310 r = pr0[5]; s = ps0[5]+z; 311 for (i = 4; i >= 0; i--) { 312 r = z*r + pr0[i]; 313 s = z*s + ps0[i]; 314 } 315 return (r/s); 316 } 317 318 319 static const GENERIC qr0[6] = { 320 0.3322091340985722351859704442e5, 321 0.8514516067533570196555001171e5, 322 0.6617883658127083517939992166e5, 323 0.1849426287322386679652009819e5, 324 0.1706375429020768002061283546e4, 325 0.3526513384663603218592175580e2, 326 }; 327 static const GENERIC qs0[6] = { 328 0.7087128194102874357377502472e6, 329 0.1819458042243997298924553839e7, 330 0.1419460669603720892855755253e7, 331 0.4002944358226697511708610813e6, 332 0.3789022974577220264142952256e5, 333 0.8638367769604990967475517183e3, 334 }; 335 336 static GENERIC 337 qone(GENERIC x) { 338 GENERIC s, r, t, z; 339 int i; 340 if (x > huge) 341 return (0.375/x); 342 343 t = 8.0/x; z = t*t; 344 /* assume x > 8 */ 345 r = qr0[5]; s = qs0[5]+z; 346 for (i = 4; i >= 0; i--) { 347 r = z*r + qr0[i]; 348 s = z*s + qs0[i]; 349 } 350 return (t*(r/s)); 351 } 352